About Clock
Two ideas cover the whole topic. The first is the angle between the hands, which comes from their positions at a given time. The second is the rate at which the minute hand gains on the hour hand, which decides how often the two line up.
What you need to understand
- The minute hand sweeps 6 degrees in a minute, because a full turn of 360 degrees takes 60 minutes.
- The hour hand sweeps 0.5 degrees in a minute, because it covers 30 degrees in an hour.
- At H o clock the hour hand stands at 30H degrees, and it moves on by half a degree for every minute that passes.
- The angle between the hands is the difference of their two positions, folded into the range 0 to 180 degrees.
- The minute hand gains 5.5 degrees a minute on the hour hand, so the hands line up 11 times in 12 hours.
- A clock that gains or loses time does so at a steady rate, so those questions are ordinary proportion.
Formulas to remember
How to work through these questions
- Write both hand positions in degrees before subtracting anything.
- Take the difference, and if it exceeds 180 degrees subtract it from 360 to get the smaller angle.
- For the counting questions, remember the hands do not line up once an hour, because the hour hand keeps moving.
- For a clock that gains or loses, reduce the problem to a rate per hour and scale it.
- Check that an angle answer lies between 0 and 180 degrees.
Mistakes that cost marks
- Forgetting that the hour hand moves between the hour marks, which shifts the angle by as much as 30 degrees.
- Giving the reflex angle when the smaller angle was asked for.
- Assuming the hands meet once every hour, which gives 24 in a day instead of 22.
- Using 6 degrees a minute for the hour hand as well as for the minute hand.
- Confusing the gain per hour with the gain over the whole period.
Worked example
What is the angle between the hands of a clock at 3:40?
- Hour hand position = 30 x 3 + 0.5 x 40 = 90 + 20 = 110 degrees.
- Minute hand position = 6 x 40 = 240 degrees.
- The difference is 240 - 110 = 130 degrees, which is already less than 180.
- So the angle between the hands is 130 degrees.
Answer: 130 degrees
Practice questions with answers
A few Clock questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
What is the angle between the hands of a clock at 11:13?
-
A
252
-
B
101.5
-
C
41.5
-
D
258.5
Answer: Option B — with explanation
Hour hand position = 30H + 0.5M degrees, minute hand position = 6M degrees.
At 11:13: hour hand = 30 x 11 + 0.5 x 13 = 673/2 degrees, minute hand = 6 x 13 = 78 degrees.
Difference = 517/2 degrees, and the smaller angle between the hands is 203/2 degrees.
Common mistakes
- added the two positions instead of finding the difference: 41.5
- gave the reflex angle instead of the smaller angle between the hands: 258.5
- treated the hour hand as if it stood still on the hour: 252
Question 2
A clock gains 6 seconds in 3 hours. How many seconds will it gain in 24 hours?
Answer: Option C — with explanation
In 3 hours it gains 6 seconds, so the gain per hour is 6 / 3 = 2 seconds.
In 24 hours the gain is 2 x 24 = 48 seconds.
Common mistakes
- added one more interval of gain than the period contains: 54
- halved the period as though the clock gained every two hours: 24
- doubled the gain: 96
Question 3
Find the angle between the hour hand and the minute hand at 5:02.
Answer: Option C — with explanation
Hour hand position = 30H + 0.5M degrees, minute hand position = 6M degrees.
At 5:02: hour hand = 30 x 5 + 0.5 x 2 = 151 degrees, minute hand = 6 x 2 = 12 degrees.
Difference = 139 degrees, and the smaller angle between the hands is 139 degrees.
Common mistakes
- added the two positions instead of finding the difference: 161
- gave the reflex angle instead of the smaller angle between the hands: 221
- treated the hour hand as if it stood still on the hour: 138
Question 4
Find the angle between the hour hand and the minute hand at 4:41.
-
A
126
-
B
254.5
-
C
345.5
-
D
105.5
Answer: Option D — with explanation
Hour hand position = 30H + 0.5M degrees, minute hand position = 6M degrees.
At 4:41: hour hand = 30 x 4 + 0.5 x 41 = 281/2 degrees, minute hand = 6 x 41 = 246 degrees.
Difference = 211/2 degrees, and the smaller angle between the hands is 211/2 degrees.
Common mistakes
- added the two positions instead of finding the difference: 345.5
- treated the hour hand as if it stood still on the hour: 126
- gave the reflex angle instead of the smaller angle between the hands: 254.5
Question 5
Find the angle between the hour hand and the minute hand at 5:16.
Answer: Option A — with explanation
Hour hand position = 30H + 0.5M degrees, minute hand position = 6M degrees.
At 5:16: hour hand = 30 x 5 + 0.5 x 16 = 158 degrees, minute hand = 6 x 16 = 96 degrees.
Difference = 62 degrees, and the smaller angle between the hands is 62 degrees.
Common mistakes
- added the two positions instead of finding the difference: 238
- gave the reflex angle instead of the smaller angle between the hands: 298
- treated the hour hand as if it stood still on the hour: 54
Frequently asked questions
Why does the hour hand move 0.5 degrees a minute?
Because it covers 30 degrees in 60 minutes, and 30 divided by 60 is 0.5. Ignoring this movement is the commonest mistake in the topic.
How many times a day do the hands of a clock coincide?
Twenty two times. They coincide 11 times in every 12 hours rather than 12, because the hour hand moves on while the minute hand is chasing it.
Can the angle between the hands be more than 180 degrees?
The two hands always define two angles that add up to 360 degrees. Questions normally ask for the smaller one, so the answer is at most 180 degrees.
How do I handle a clock that gains time?
Work out the gain per hour and multiply by the number of hours. The rate is steady, so it is a straightforward proportion.
